Ch1.Analog radar signal design and digital signal processing

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1、ChapteriAnalogradarsignaldesignanddigitalsignalprocessing--aHeisenbergnilpotentLiegroupapproachbyWALTERSCHEMPPIneverymathematicalinve~tigation~thequestionwillarisewhetherwecanapplyourmathematicalresultstotherealworld.V.I.ARNOLD(1983)Wewillciteinstaneeeofpureandappliedmathematiciansdoingt

2、hesameoranalogousmathematics,butbecauseo/thelacko/communicationneitherknewoftheothers'work.L.AUSLANDER(1979)ABSTRACT:Thenotionsofanaloganddigitalradarauto-andcross-ambiguityfunc-tionsareontheborderlinewithmathematics,physics,andelectricalengineering.Thispaperpresentsthesolutionsoftwoprob

3、lemsofanalogradarsignaldesign:thesynthesisproblem(posedin1953)andtheinwarianceproblemforambiguitysurfacesoverthesympleetictime-frequencyplane.Bothsolutionsareachievedviaharmonicanalysisonthedifferentialprincipalfiberbundleoverthetwo-dimensionalpolarized(resp.isotropic)cross-sectionwithst

4、ructuregroupisomorphictotheone-dimensionalcenterofthesimplyconnectedrealHeisenbergnilpotentLiegroup.Inthisway,thelinearoscillatorrepre-sentationofthethree-dimensionalrealmetaplecticgroupgivesrisetoaprocedureforgeneratingtheenergy-preservinglinearautomorphismsofanygivenradarambiguitysur-f

5、aceoverthetime-frequencyplanebymeansofchirpwaveforms(linearfrequencymodu-latedsignals).Inthefieldofdigitalsignalprocessing,theWhittaker-Shannon-Kotel'nikovsamplingtheoremalsofitstheframeworkofnilpotentharmonicanalysis.ThebasicideaistorealizethelinearSchriidingerrepresentationbythelinearl

6、atticerepresentationactinginacomplexHilbertspacemodeledonthecompactHeisenbergnilmanifold,towit,thedifferentialprincipalfiberbundleoverthetwo-dimensionalcompacttorus~withstructuregroupisomorphictotheone-dimensionalcenterofthereducedrealHeisenbergnilpotentLiegroup.Inthesameveinwelookuponth

7、efiniteFouriertransform,andfinally,basedontheambiguitysurfaceconservationprinciple,thepaperdealsinageometricwaywiththephasediscontinuityofFourierandmicrowaveoptics.Itfollowsthatanaloganddigitalsignalprocessing,aswellasFourieroptics,haveadeepgeometriccommonrootinnilpotenth